Theorem ssrabdv 3226

Description: Subclass of a restricted class abstraction (deduction form). (Contributed by NM, 31-Aug-2006.)

Hypotheses

Ref	Expression
ssrabdv.1	|- ( ph -> B C_ A )
ssrabdv.2	|- ( ( ph /\ x e. B ) -> ps )

Assertion

Ref	Expression
ssrabdv	|- ( ph -> B C_ { x e. A | ps } )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem ssrabdv

Step	Hyp	Ref	Expression
1		ssrabdv.1	. 2 |- ( ph -> B C_ A )
2		ssrabdv.2	. . 3 |- ( ( ph /\ x e. B ) -> ps )
3	2	ralrimiva 2543	. 2 |- ( ph -> A. x e. B ps )
4		ssrab 3225	. 2 |- ( B C_ { x e. A | ps } <-> ( B C_ A /\ A. x e. B ps ) )
5	1, 3, 4	sylanbrc 415	1 |- ( ph -> B C_ { x e. A | ps } )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   {crab 2452   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-in 3127  df-ss 3134

This theorem is referenced by: (None)